Contraction Analysis of Time-Delayed
Communications and Group Cooperation
Wei Wang
Jean-Jacques E. Slotine
Nonlinear Systems Laboratory, MIT
Cambridge, Massachusetts, 02139
Email: wangwei@mit.edu
Nonlinear Systems Laboratory, MIT
Cambridge, Massachusetts, 02139
Email: jjs@mit.edu
Abstract— We study stability of interacting nonlinear systems
with time-delayed communications, using contraction theory
and a simplified wave variable design inspired by robotic
teleoperation. We show that contraction is preserved through
specific time-delayed feedback communications, and that this
property is independent of the values of the delays. The
approach is then applied to group cooperation with linear
protocols, where it is shown that synchronization can be made
robust to arbitrary delays.
NSL-041001, October 2004, revised December, 2005
To be published in I.E.E.E. Trans. Automatic Control
I. I NTRODUCTION
In many engineering applications, communications delays
between subsystems cannot be neglected. Such an example
is bilateral teleoperation, where signals can experience significant transmission delays between local and remote sites.
Throughout the last decade, both internet and wireless technologies have vastly extended practical communication distances. Information exchange and cooperation can now occur
in very widely distributed systems, making the effect of time
delays even more central.
In the context of telerobotics, [2] proposed a control law
for force-reflecting teleoperators which preserves passivity,
and thus overcomes the instability caused by time delays.
The idea was reformulated in [18] in terms of scattering
or “wave” variables [6], [3]. Transmission of wave variables across communication channels ensures stability without
knowledge of the time delay. Further extensions to internetbased applications were developed [16], [17], [4], in which
communication delays are variable.
Recently, [14], [23] extended the application of wave variables to a more general context by performing a nonlinear
contraction analysis [13], [14], [1], [7] of the effect of timedelayed communications between contracting systems. This
paper modifies the design of the wave variables proposed
in [14], [23]. Specifically, a simplified form provides an
effective analysis tool for interacting nonlinear systems with
time-delayed feedback communications. For appropriate coupling terms, contraction as a generalized stability property is
preserved regardless of the delay values. This also sheds a new
light on the well-known fact in bilateral teleoperation that even
small time-delays in feedback Proportional-Derivative (PD)
controllers may create stability problems for simple coupled
second-order systems, which in turn motivated approaches
based on passivity and wave variables [2], [18]. The approach
is then applied to derive the paper’s main results on the group
cooperation problem with delayed communications. We show
that synchronization with linear protocols [19], [15] is robust
to time delays and network connectivity without requiring
the delays to be known or equal in all links. In a leaderless
network, all the coupled elements reach a common state which
depends on the initial conditions and the time delays, while
in a leader-followers network the group agreement point is
fixed by the leader. The approach is suitable to study both
continuous-time and discrete-time models.
II. C ONTRACTION A NALYSIS OF T IME -D ELAYED
C OMMUNICATIONS
This introductory section shows that a simplified form of
the transmitted wave variables in [14] can be applied to
analyze time-delayed feedback communications. The results
which follow could be obtained using a variety of alternative
techniques. For instance, they could be easily derived based on
input-output analysis [31], although in an L2 sense rather than
exponentially. The use of modified wave variables provides a
unifying framework which will be central in the derivation of
the paper’s main results in section III.
Consider two interacting systems of possibly different dimensions,
(
ẋ1 = f1 (x1 , t) + G21 τ 21
(1)
ẋ2 = f2 (x2 , t) + G12 τ 12
where x1 ∈ Rn1 , x2 ∈ Rn2 , τ 12 , τ 21 ∈ Rn , and G12 ∈
Rn1 ×n , G21 ∈ Rn2 ×n are two constant matrices. Inputs
τ ij are computed by transmitting between the two systems
simplified “wave” variables, defined as
u21
u12
= GT21 x1 + k21 τ 21
= GT12 x2 + k12 τ 12
v12 = GT21 x1
v21 = GT12 x2
where k12 and k21 are two strictly positive constants. Because
of time delays, one has
u12 (t) = v12 (t − T12 )
u21 (t) = v21 (t − T21 )
where T12 and T21 are two positive constants. Note that
subscripts containing two numbers indicate the communication
direction, e.g., subscript “12” refers to communication from
node 1 to 2. This notation will be helpful in Section III, where
results will be extended to groups of interacting subsystems.
Consider, similarly to [14], [23], the differential length
k21
k12
1
δxT1 δx1 +
δxT2 δx2 +
V1,2
2
2
2
V =
where
V1,2 =
Z
t
t−T12
T
δv12
δv12 dǫ +
Z
t
t−T21
T
δv21
δv21 dǫ
This yields
∂f1
∂f2
δx1 + k12 δxT2
δx2
∂x1
∂x2
2
k2
k21
δτ T21 δτ 21 − 12 δτ T12 δτ 12
−
2
2
If f1 and f2 are both contracting with identity metrics (i.e., if
∂f2
∂f1
∂x1 and ∂x2 are both uniformly negative definite), then V̇ ≤
0, and V is bounded and tends to a limit. Applying Barbalat’s
lemma [22] in turn shows that if V̈ is bounded, then V̇ tends
to zero asymptotically, which implies that δx1 , δx2 , δτ 12 and
δτ 21 all tend to zero. Regardless of the values of the delays,
all solutions of system (1) converge to a single trajectory,
independent of the initial conditions. In the sequel we shall
assume that V̈ can indeed be bounded as a consequence of
the boundedness of V .
This result has a useful interpretation. Expanding system
dynamics (1) yields
(
ẋ1 = f1 (x1 , t) + k121 G21 (GT12 x2 (t − T21 ) − GT21 x1 (t))
ẋ2 = f2 (x2 , t) + k112 G12 (GT21 x1 (t − T12 ) − GT12 x2 (t))
V̇
=
k21 δxT1
If we assume further that x1 and x2 have the same dimension,
and choose G12 = G21 = G, the whole system is actually
equivalent to two diffusively coupled subsystems
(
ẋ1 = f1 (x1 , t) + k121 GGT ( x2 (t − T21 ) − x1 (t) )
ẋ2 = f2 (x2 , t) + k112 GGT ( x1 (t − T12 ) − x2 (t) )
This implies that, for appropriate coupling terms, contraction
as a generalized stability property will be preserved regardless
of the time delays and the delay values.
This result does not contradict the well-known fact in
teleoperation that even small time delays in bilateral PD
controllers may create stability problems for coupled secondorder systems [2], [18], [17], [4], which motivates approaches
based on passivity and wave variables. In fact, a key condition
for contraction to be preserved is that the coupling gains be
symmetric positive semi-definite in the same metric as the
subsystems.
Example 2.1 Consider two identical second-order systems
coupled through time-delayed feedback PD controllers
(
h1 = kd (ẋ2 (t − T21 ) − ẋ1 (t)) + kp (x2 (t − T21 ) − x1 (t))
h2 = kd (ẋ1 (t − T12 ) − ẋ2 (t)) + kp (x1 (t − T12 ) − x2 (t))
where h1 = ẍ1 + bẋ1 + ω 2 x1 , h2 = ẍ2 + bẋ2 + ω 2 x2 , and
b > 0, ω > 0. If T12 = T21 = 0, x1 and x2 converge together
exponentially regardless of initial conditions, which makes the
origin a stable equilibrium point. If T12 , T21 > 0, a simple
coordinate transformation yields
x2 (t − T21 )
x1
ωy1 − bx1
ẋ1
)
+ K(
−
=
y1
y2 (t − T21 )
−ωx1
ẏ1
x1 (t − T12 )
x2
ωy2 − bx2
ẋ2
)
+ K(
−
=
y2
y1 (t − T12 )
−ωx2
ẏ2
ωy2 − bx2
ωy1 − bx1
are
and f2 =
where f1 =
−ωx2
−ωx1
both (semi-)contracting with identity
However, the
metric [29].
kd 0
is neither symtransformed coupling gain K =
kp
0
ω
metric nor positive semi-definite for any kp 6= 0. Contraction
cannot be preserved in this case, and the coupled systems turn
out to be unstable for large enough delays as the simulation
result in Figure 1(a) illustrates. While in Figure 1(b), once we
set kp = 0, the overall system is therefore contracting. 2
60
(a). PD Control
x
1
x
2
8
(b). D Control
x
1
x
6
2
40
4
20
2
0
0
−2
−20
−4
−6
−40
−8
−60
0
20
40
t(sec)
60
−10
0
20
40
t(sec)
60
Fig. 1. Simulation results of two coupled mass-spring-damper systems with
(a) PD control and (b) D control. Parameters are b = 0.5, ω 2 = 5, T12 = 2s,
T21 = 4s, kd = 1, kp = 5 in (a) and kp = 0 in (b). Initial conditions, chosen
randomly, are identical for the two plots.
The instability mechanism in the above example is actually
very similar to that of the classical Smale model [26], [29]
of spontaneous oscillation, in which two or more identical
biological cells, inert by themselves, tend to self-excited
oscillations through diffusion interactions. In both cases, the
instability is caused by a non-identity metric, which makes
the transformed coupling gains lose positive semi-definiteness.
Note that the relative simplicity with which both phenomena
can be interpreted makes fundamental use of the notion of a
metric, central to contraction theory.
Finally, it is straightforward to show that the results here
apply recursively to feedback hierarchies of such systems.
III. G ROUP C OOPERATION WITH T IME -D ELAYED
C OMMUNICATIONS
We now present the paper’s main results. Recently, synchronization or group agreement has been the object of extensive
literature [11], [12], [19], [20], [21], [27], [28]. Understanding
natural aggregate motions as in bird flocks, fish schools, or
animal herds may help achieve desired collective behaviors
in artificial multi-agent systems. In our previous work [24],
[29], a synchronization condition was obtained for a group of
coupled nonlinear systems, where the number of the elements
can be arbitrary and the network structure can be very general.
In this section, we study a simplified continuous-time model
of schooling or flocking with time-delayed communications,
and generalize recent results in the literature [19], [15]. In
particular, we show that synchronization is robust to time
delays both for the leaderless case and for the leader-followers
case, without requiring the delays to be known or equal in all
links. Similar results are then derived for discrete-time models.
A. Leaderless Group
We first investigate a flocking model without group leader.
The dynamics of the ith element is given as
X
ẋi =
Kji (xj − xi )
(2)
j∈Ni
where i = 1, . . . , n and xi ∈ Rm . Ni denotes the set of
the active neighbors of element i, which for instance can be
defined as the set of the nearest neighbors within a certain
distance around i; and Kji is the coupling gain, which is
assumed to be symmetric and positive definite.
Theorem 1: Consider n coupled elements with linear protocol (2). The whole system will tend to reach a group agreement
x1 (t) = · · · = xn (t) = n1 (x1 (0) + · · · + xn (0)) exponentially
if the network is connected, and the coupling links are either
bidirectional with Kji = Kij , or unidirectional but formed in
closed rings with identical gains.
Theorem 1 is derived in [24], [29] based on partial contraction analysis, and the result can be extended further to
time-varying couplings (Kji = Kji (t)), switching networks
(Ni = Ni (t)) and looser connectivity conditions.
Assume now that time delays are non-negligible in communications. The dynamics of the ith element turns out to be
X
ẋi =
Kji ( xj (t − Tji ) − xi (t) )
(3)
j∈Ni
Theorem 2: Consider n coupled elements (3) with timedelayed communications. Regardless of the explicit values
of the delays, the whole system will tend to reach a group
agreement x1 (t) = · · · = xn (t) asymptotically if the network
is connected, and the coupling links are either bidirectional
with Kji = Kij , or unidirectional but formed in closed rings
with identical gains.
Proof: For notational simplicity, we first assume that all the
links are bidirectional with Kji = Kij , but the time delays
could be different along opposite directions, i.e., Tji 6= Tij .
Thus, Equation (3) can be transformed to
X
ẋi =
Gji τ ji
j∈Ni
where τ ji and correspondingly τ ij are defined through
uji
uij
= GTji xi + τ ji
=
GTij xj
+ τ ij
vij = GTji xi
vji =
(4)
GTij xj
with Gij = Gji > 0 and Kji = Kij = Gij GTij . Define
n
1 X
1X T
δxi δxi +
V =
2 i=1
2
(i,j)∈N
Vi,j
(5)
where N = ∪ni=1 Ni denotes the set of all active links, and
Vi,j is defined as in Section II for each link connecting two
nodes i and j. Therefore
1 X
(δτ Tji δτ ji + δτ Tij δτ ij )
V̇ = −
2
(i,j)∈N
Since V̇ (t) is non-positive, V (t) is bounded if initial states
are bounded. One can easily show that δxi (t) is bounded for
any i. This implies that δτ ij (t) and δ ẋi (t) are both bounded,
and so is δ τ̇ ij (t) since
δ τ̇ ji (t) = GTij δ ẋj (t − Tji ) − GTji δ ẋi
Thus we can say that V̈ (t) is bounded. According to Barbalat’s
lemma, V̇ will then tend to zero asymptotically, which implies
that, ∀(i, j) ∈ N , δτ ji and δτ ij tend to zero asymptotically.
Thus we know that ∀i, δ ẋi tends to zero. Now in general, a
vanishing δ ẋi does not necessarily imply that δxi is convergent. However it does in this case, because otherwise it would
contradict the fact that δxi tends to be periodic with constant
period Tji + Tij ,
δuji (t)
= GTji δxi (t) + δτ ji (t) = GTij δxj (t − Tji )
δuij (t)
= GTij δxj (t) + δτ ij (t) = GTji δxi (t − Tij )
We can also conclude that, if ∀i, δxi is convergent, they will
tend to a steady state
δx1 (t) = · · · = δxn (t) = c
where c is a constant vector whose value depends on the
specific trajectories we analyze. Moreover, we notice that, in
the state-space, any point inside the region x1 = · · · = xn is
invariant to (3). By path integration this implies immediately
that regardless of the delay values or the initial conditions, all
solutions of system (3) will tend to reach a group agreement
x1 = · · · = xn asymptotically.
In the case that coupling links are unidirectional but form
closed rings with identical coupling gains in each ring, we set
Z t
n
1 X
1X T
T
δxi δxi +
δvji
δvji dǫ
V =
2 i=1
2
t−Tji
(j→i)∈N
and the rest of the proof is the same. The case when both
types of links are involved is similar.
2
Example 3.1 Compared with Theorem 1, the group agreement point in Theorem 2 generally does not equal the average
value of the initial conditions, but depends on the values of
the time delays.
Consider the cooperative group (3) with one-dimensional
xi , n = 6, and a two-way chain structure
1o
/2o
/3o
/4o
/5o
/6
The coupling gains are set to be identical with k = 5. The
delay values are different, and each is chosen randomly around
0.5 second. Simulation results are plotted in Figure 2.
2
(a). Without Delays
25
(b). With Delays
25
xi
20
20
15
15
10
10
5
5
0
0
−5
−5
−10
−10
−15
are bidirectional, we can transform the equation (6) to
X
ẋi =
Gji τ ji + γi K0i (x0 − xi )
xi
j∈Ni
−15
−20
−20
t
t
−25
0
5
10
−25
0
5
10
15
20
where τ ji and τ ij are defined the same as those in (4).
Considering the same Lyapunov function V as (5), we get
n
X
1 X
(δτ Tji δτ ji +δτ Tij δτ ij )
γi δxTi K0i δxi −
V̇ = −
2
i=1
(i,j)∈N
Fig. 2. Simulation results for Example 3.1 without delays and with delays.
Initial conditions, chosen randomly, are the same for each simulation. Group
agreement is reached in both cases, although the agreement value is different.
Note that the conditions on coupling gains can be relaxed.
If the links are bidirectional, we do not have to require Kij =
Kji . Instead, the dynamics of the ith element could be
X
( xj (t − Tji ) − xi (t) )
ẋi = Ki
j∈Ni
where Ki = k1i GGT and G is unique through the whole
network. The proof is the same except that we incorporate
ki into the wave variables and the function V . Such a design
brings more flexibility to cooperation-law design. The discretetime model studied in Section III-C is in this spirit. A similar
condition was derived in [5] for a delayless swarm model.
Model (2) with delayed communications was also studied
in [19], but the result is limited by the assumptions that
communication delays are equal in all links and that the selfresponse part in each coupling uses the same time delay.
Recently, [15] independently analyzed system (3) in the scalar
case with the assumption that delays are equal in all links.
∪ni=1 Ni
where N =
denotes the set of all active links among
the followers. Applying Barbalat’s lemma shows that V̇ will
tend to zero asymptotically. It implies that ∀i, if γi = 1, δxi
will tend to zero, as well as δτ ji and δτ ij ∀(i, j) ∈ N .
Moreover, since
δτ ji (t) = GTji ( δxj (t − Tji ) − δxi (t) )
we conclude that if the whole leader-followers network is
connected, the virtual dynamics will converge to δx1 (t) =
· · · = δxn (t) = 0 regardless of the initial conditions or
the delay values, i.e., the whole system is asymptotically
contracting. All solutions will converge to a particular one,
which in this case is the point x1 (t) = · · · = xn (t) = x0 . The
proof is similar for unidirectional links in closed rings.
2
Example 3.2 Consider a leader-followers network (6) with
one-dimensional xi , n = 6, and structured by
/2o
/1o
/3o
/4o
/5o
/6
0
The state of the leader is constant with value x0 = 10. All the
coupling gains are set to be identical with k = 5. The delays
are not equal, each of which is chosen randomly around 0.5
second. Simulation results are plotted in Fig. 3.
2
(a). Without Delays
30
xi
i
B. Leader-Followers Group
Similar analysis can be applied to study coupled networks
with group leaders. Consider such a model
X
ẋi =
Kji (xj (t − Tji ) − xi (t)) + γi K0i (x0 − xi ) (6)
j∈Ni
where i = 1, . . . , n; x0 is the state of the leader, which we
first assume to be a constant; xi are the states of the followers;
Ni indicate the neighborship among the followers; and γi = 0
or 1 represents the unidirectional links from the leader to the
corresponding followers. For each non-zero γi , the coupling
gain K0i is positive definite.
Theorem 3: Consider a leader-followers network (6) with
time-delayed communications. Regardless of the explicit values of the delays, the whole system will tend to reach a group
agreement x1 (t) = · · · = xn (t) = x0 asymptotically if the
whole network is connected, and the coupling links among
the followers are either bidirectional with Kji = Kij , or
unidirectional but formed in closed rings with identical gains.
Proof: Exponential convergence of the leader-followers network (6) without delays has been shown in [24], [29] using
contraction theory. If the communication delays are nonnegligible, and assuming that all the links among the followers
25
20
20
15
15
10
10
5
5
0
0
−5
−5
−10
−15
0
(b). With Delays
30
x
25
−10
t
t
5
10
15
20
−15
0
10
20
30
40
50
Fig. 3. Simulation results for Example 3.2 without delays and with delays.
Initial conditions, chosen randomly, are the same for each simulation. In both
cases, group agreement to the leader value x0 is reached.
Note that even if x0 is not a constant, i.e., the dynamics of
the ith element is given as
X
ẋi =
Kji (xj (t−Tji )−xi (t))+γi K0i (x0 (t−T0i )−xi (t))
j∈Ni
the whole system is still asymptotically contracting according
to exactly the same proof. Regardless of the initial conditions,
all solutions converge to a particular one, which in this case
depends on the dynamics of x0 and the explicit values of
the delays. Moreover, if x0 is periodic, as one of the main
properties of contraction [13], all the followers’ states xi will
tend to be periodic with the same period as x0 .
C. Discrete-Time Models
Simplified wave variables can also be applied to study timedelayed communications in discrete-time models. Consider the
model of flocking or schooling studied in [11], [28]:
X
1
xi (t + 1) = xi (t) +
( xj (t) − xi (t) )
1 + ni
j∈Ni
where i = 1, . . . , n, and ni equals the number of the neighbors
of element i. As proved in [11], the whole system will tend
to reach a group agreement if the network is connected in a
very loose sense.
Assume now that time delays are non-negligible in communications. The update law of the ith element changes to
X
1
xi (t+1) = xi (t) +
( xj (t−Tji ) − xi (t) ) (7)
1 + ni
j∈Ni
where the delay value Tji is an integer based on the number of
updating steps. As in previous sections, Tji could be different
for different communication links, or even different along
opposite directions on the same link.
We assume: all elements update their states synchronously;
the time interval between any two updating steps is a constant;
the network structure is connected and fixed, i.e., ni is a
positive integer ∀i; and the value of the neighborhood radius
r is unique throughout the whole network, which leads to the
fact that all interactions are bidirectional.
Theorem 4: Consider n coupled elements (7) with timedelayed communications. Regardless of the explicit values
of the delays, the whole system will tend to reach a group
agreement x1 (t) = · · · = xn (t) asymptotically. P
Proof: See [30], with xi (t + 1) = xi (t) +
j∈Ni τji (t)
and wave variables uji = xi + (1 + ni ) τji , vij = xi .
A similar analysis leads to the same result as Theorem 3 for
a discrete-time model with a leader-followers structure, both
of which can be applied to study group cooperation problem
with asynchronous updating instants [8].
IV. C ONCLUSION
This paper introduces modified wave variables in the context
of contraction analysis, and shows that they yield effective
and simple tools for analyzing interacting systems and group
synchronization with time-delayed feedback communications.
Future work includes coupled networks with switching topologies and time-varying time-delays.
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